## MFXXXX

MF2050 Discrete Time Financial Models

Credit Weighting: 5

Teaching Period(s): Teaching Period 1.

No. of Students: Min 5, Max 30.

Pre-requisite(s): MA1054, MA1057, MA1058, AM1053, ST1051

Co-requisite(s): ST2054

Teaching Methods: 24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

Module Co-ordinator: Professor Bernard Hanzon, School of Mathematical Sciences.

Lecturer(s): Dr Grigory Temnov, School of Mathematical Sciences.

Module Objective: Introduction to the theory and application of discrete-time financial models.

Module Content: Discrete time Finite state space models: Application of discrete time Martingales; Introduction to risk neutral measures; Binomial Option pricing models; Arrow securities, Pricing in incomplete markets.

Learning Outcomes: On successful completion of this module, students should be able to:
· Construct binomial tree model, determine the associated risk-neutral probability;
· Describe the basis concepts of investment strategy analysis, including self-financing, arbitrage free, admissible, predictable investment strategies; to apply these concepts to the binomial tree model;
· Price forward contracts and perform price calculations for stocks without and with dividend payments; to price futures contracts, to do cash-flow calculations involving futures;
· Calculate the price of various European options in a binomial tree model;
· Explain the First and Second Fundamental Theorem of Asset Pricing in the One-Period model by using matrix calculations and linear algebra and apply them to basic derivative securities;
· Solve problems involving complete markets, incomplete One-Period markets and hedging of basic derivative securities;
· Construct and apply the Cox-Ross-Rubinstein model;
· Calculate the price of various European options in a binomial tree model; construct a replicating portfolio in a binomial tree model;
· Construct a replicating portfolio in a binomial tree model; Perform calculations with the Black-Scholes call option price formula and be able to prove various steps in the derivation of the Black-Scholes call option price formula.

Assessment: Total Marks 100: End of Year Written Examination 80 marks; Continuous Assessment 20 marks (one in-class test: 10 marks; Homework assignments 10 marks (3 x 3.33 marks each)).

Compulsory Elements: End of Year Written Examination; Continuous Assessment.

Penalties (for late submission of Course/Project Work etc.): Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

Pass Standard and any Special Requirements for Passing Module: 40%.

End of Year Written Examination Profile: 1 x 1½ hr(s) paper(s).

Requirements for Supplemental Examination: 1 x 1½ hr(s) paper(s) to be taken in Autumn. The mark for Continuous Assessment is carried forward.

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MF2052 Introduction to Financial Mathematics

Credit Weighting: 10

Teaching Period(s): Teaching Periods 1 and 2.

No. of Students: Min 5, Max 40.

Pre-requisite(s): ST1051, MA1054, or equivalent

Co-requisite(s): None

Teaching Methods: 48 x 1hr(s) Lectures; Other (20 x 1hr Tutorials/Practicals).

Module Co-ordinator: Ms Linda Daly, Department of Statistics.

Lecturer(s): Mr Damian Conway, Department of Statistics.

Module Objective: To provide an introduction to financial and actuarial mathematics

Module Content: Theory of interest rates; Annuities; Discounted cash flows; Equations of value; Analysis of loan schedules; Valuation of fixed interest securities and investment performance; Arbitrage and forward contracts; Term structure of interest rates; Stochastic interest rate models; Project appraisal.

Learning Outcomes: On successful completion of this module, students should be able to:
· Apply a generalised cashflow model to analyse financial transactions.
· Describe how to take into account the time value of money using the concepts of compound interest and discounting.
· Carry out calculations involving the present value and the accumulated value of a stream of payments using specified rates of interest.
· Define an equation of value and apply this to calculate loan repayments under a Repayment Mortgage contract.
· Analyse and compare investment projects in terms of their Net Present Value and Discounted Payback Period.
· Describe the investment and risk characteristics of the following types of assets: fixed interest government borrowings, fixed interest borrowing by other bodies, shares and other equity-type finance and derivatives.
· Analyse elementary compound interest problems allowing for both income / capital gains tax-liabilities and calculate the real yield obtained by an investor on a number of bond-type transactions.
· Define the concept of arbitrage, explain the significance of the No Arbitrage assumption and use this assumption to calculate the forward price of a number of derivative-type contracts.
· Evaluate the duration and convexity of a cashflow sequence and explain how such concepts are used in the (Redington) immunisation of a portfolio of liabilities.

Assessment: Total Marks 200: End of Year Written Examination 160 marks; Continuous Assessment 40 marks (2 x in-class tests 12 marks each, 2 x homework 8 marks each).

Compulsory Elements: End of Year Written Examination; Continuous Assessment.

Penalties (for late submission of Course/Project Work etc.): Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

Pass Standard and any Special Requirements for Passing Module: 40%.

End of Year Written Examination Profile: 1 x 3 hr(s) paper(s).

Requirements for Supplemental Examination: 1 x 3 hr(s) paper(s) to be taken in Autumn. The mark for Continuous Assessment is carried forward.

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MF2053 Financial Modelling for Actuarial Science 1

Credit Weighting: 5

Teaching Period(s): Teaching Period 1.

No. of Students: Min 5, Max 20.

Pre-requisite(s): None.

Co-requisite(s): MF2052, ST2054

Teaching Methods: 24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

Module Co-ordinator: Mr Damian Conway, Department of Statistics.

Lecturer(s): Mr Damian Conway, Department of Statistics, Invited members of the Actuarial Profession.

Module Objective: To provide an introduction to the key elements and techniques of corporate finance as used by actuaries and quantitative financial analysts.

Module Content: Risks Sensitivities (the Greeks); Derivatives & Option Contracts; Efficient Market Hypothesis; Utility Theory; Investment Risk; Asset Pricing Models; CAPM; Portfolio Theory.

Learning Outcomes: On successful completion of this module, students should be able to:
· Describe methods by which derivatives & options may be valued.
· Define and carry out calculations using the Greeks.
· Describe the various forms of the efficient market hypothesis and discuss its validity.
· Outline the key concepts of Mean Variance Portfolio Theory and use the theory to perform calculations in relation to a portfolio of risky assets.
· Outline the CAPM and describe the assumptions underlying the model.
· Describe the key elements of utility theory and perform calculations using utility functions in order to decide on investment strategies.
· Describe various forms of investment risk measures and perform calculations using these risk measures.
· Describe the single and multi-factor model of asset returns.

Assessment: Total Marks 100: Continuous Assessment 100 marks (2 x in-class tests (1 x 10 marks, 1 x 80 marks); 1 x homework, 10 marks).

Compulsory Elements: Continuous Assessment.

Penalties (for late submission of Course/Project Work etc.): Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

Pass Standard and any Special Requirements for Passing Module: 40%.

End of Year Written Examination Profile: No End of Year Written Examination.

Requirements for Supplemental Examination: Marks in passed element(s) of Continuous Assessment are carried forward, Failed element(s) of Continuous Assessment must be repeated.

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MF3052 Derivatives, Securities and Option Pricing

Credit Weighting: 5

Teaching Period(s): Teaching Period 1.

No. of Students: Min 5, Max 30.

Pre-requisite(s): MF2050, MF2053, MA2051, MA2054, MA2071, AM2071

Co-requisite(s): None

Teaching Methods: 24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

Module Co-ordinator: Professor Bernard Hanzon, School of Mathematical Sciences.

Lecturer(s): Professor Bernard Hanzon, School of Mathematical Sciences.

Module Objective: To develop the necessary skills to construct and apply asset liability models and to value financial derivatives

Module Content: Derivatives, securities and options; European, American and Asian options; exotic options; hedging strategies and trading mechanisms.

Learning Outcomes: On successful completion of this module, students should be able to:
· Construct pay-off functions for various financial derivatives and apply the put-call parity rule
· Apply Ito's lemma and perform calculations with the lognormal probability density function
· Apply the Black-Scholes partial differential equation for option valuation and construct boundary conditions to this for various European options; to construct the corresponding delta hedging portfolio
· Apply the Black-Scholes formulae for European call options
· Construct solutions in the form of an integral to the diffusion equation and to transform the Black-Scholes partial differential equation into a diffusion equation
· Perform calculations and analysis of the solution in the form of an integral of the Black Scholes partial differential equation
· Describe and use the linear complementarity equations for American options
· Value and analyze exotic options such as compound options. barrier options, Asian options, lookback options, Russian option and more general path-dependent options.

Assessment: Total Marks 100: End of Year Written Examination 80 marks; Continuous Assessment 20 marks (one in-class test: 10 marks; Homework assignments 10 marks (3 x 3.33 marks each)).

Compulsory Elements: End of Year Written Examination; Continuous Assessment.

Penalties (for late submission of Course/Project Work etc.): Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

Pass Standard and any Special Requirements for Passing Module: 40%.

End of Year Written Examination Profile: 1 x 1½ hr(s) paper(s).

Requirements for Supplemental Examination: 1 x 1½ hr(s) paper(s) to be taken in Autumn. The mark for Continuous Assessment is carried forward.

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MF3053 Financial Modelling for Actuarial Science 2

Credit Weighting: 5

Teaching Period(s): Teaching Period 2.

No. of Students: Min 5, Max 20.

Pre-requisite(s): MF2053, MF3052, ST3053

Co-requisite(s): None

Teaching Methods: 24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

Module Co-ordinator: Dr James Grannell, School of Mathematical Sciences.

Lecturer(s): Dr Grigory Temnov, School of Mathematical Sciences.

Module Objective: To provide an understanding of the key models and concepts used to value assets and make investment decisions.

Module Content: Brownian Motion; Martingales; Wilkie Model; Binomial Model; Ito Calculus; Black-Scholes formula; Derivatives and Options; Term Structure of Interest Rates; Credit Risk.

Learning Outcomes: On successful completion of this module, students should be able to:
· Discuss and apply the main concepts of Brownian Motion and Ito Calculus;
· Explain the concept of a martingale and evaluate simple martingales;
· Describe various models of security prices, such as the Wilkie model and other autoregressive models, and perform calculations using these models;
· Construct binomial tree models to value options and determine the associated risk neutral probability measures;
· Describe and perform calculations with the Black-Scholes derivative pricing model;
· Outline simple models of Credit Risk;
· Describe the main forms of derivative contracts;
· Price forward contracts and calculate the value of various options;
· Describe various models of the term structure of interest rates and carry out calculations with these models.

Assessment: Total Marks 100: Continuous Assessment 100 marks (2 x in-class tests (1 x 20 marks; 1 x 80 marks)).

Compulsory Elements: Continuous Assessment.

Penalties (for late submission of Course/Project Work etc.): None.

Pass Standard and any Special Requirements for Passing Module: 40%.

End of Year Written Examination Profile: No End of Year Written Examination.

Requirements for Supplemental Examination: Marks in passed element(s) of Continuous Assessment are carried forward, Failed element(s) of Continuous Assessment must be repeated.

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MF4051 Continuous Time Financial Models

Credit Weighting: 5

Teaching Period(s): Teaching Period 1.

No. of Students: Min 5, Max 30.

Pre-requisite(s): MF3052, ST3053, AM3063, MA3051, MF2050, MF3050

Co-requisite(s): MF4054

Teaching Methods: 24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

Module Co-ordinator: Professor Bernard Hanzon, School of Mathematical Sciences (Head of Department of Mathematics).

Lecturer(s): Professor Bernard Hanzon, School of Mathematical Sciences.

Module Objective: To introduce continuous-time financial models.

Module Content: Application of stochastic calculus to continuous time asset pricing models. Relation with discrete time models. The no-arbitrage principle. Derviation of Black-Scholes equations. Construction of replicating portfolios. Models for interest rates. Heath-Jarrow-Morton modifications. Credit risk modelling.

Learning Outcomes: On successful completion of this module, students should be able to:
· Define and apply various concepts related to continuous time dynamic portfolio strategies and to define, to apply the concept of an arbitrage free market and a complete market, and to derive the Black-Scholes partial differential equation using the construction of a replicating portfolio;
· Give and perform calculations with the representation of the price of a European option as an expectation under the pricing measure;
· Define and perform calculations with various representations of interest rates;
· Analyze and perform calculations with the term structure equation for short rate models and with the bond options formula under various model assumptions;
· Describe the HJM drift condition for forward rate models and to perform calculations;
· Derive specific results for options that are not model dependant;
· Show how to use binominal trees for option pricing;
· Describe the different approaches to credit risk modelling; structural models, reduced from models, intensity based models.

Assessment: Total Marks 100: End of Year Written Examination 80 marks; Continuous Assessment 20 marks (one in-class test: 10 marks; Homework assignments 10 marks (3 x 3.33 marks each)).

Compulsory Elements: End of Year Written Examination; Continuous Assessment.

Penalties (for late submission of Course/Project Work etc.): Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

Pass Standard and any Special Requirements for Passing Module: 40%.

End of Year Written Examination Profile: 1 x 1½ hr(s) paper(s).

Requirements for Supplemental Examination: 1 x 1½ hr(s) paper(s) to be taken in Autumn. The mark for Continuous Assessment is carried forward.

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MF4052 Computational Finance

Credit Weighting: 5

Teaching Period(s): Teaching Period 2.

No. of Students: Min 5, Max 30.

Pre-requisite(s): MF3052, AM2060, MA3051, MA2055

Co-requisite(s): None

Teaching Methods: 24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials; 6 x 1hr(s) Practicals.

Module Co-ordinator: Professor Bernard Hanzon, School of Mathematical Sciences.

Lecturer(s): Professor Bernard Hanzon, School of Mathematical Sciences.

Module Objective: To impart intensive training on applied computational techniques used in Finance.

Module Content: Binomial tree pricing methods; numerical methods for partial differential equations arising in financial applications; moving boundaries and American options; Monte Carlo simulation; applied financial time series techniques; use of software packages.

Learning Outcomes: On successful completion of this module, students should be able to:
· Construct binomial trees that correspond to a geometric Brownian motion model, to calculate European and American options prices using the binomial tree method and to write computer programs for that;
· Calculate European and American option prices using finite difference methods and to write computer programs for that;
· Perform a stability analysis for such schemes using matrix methods;
· Calculate European and American option prices using Monte Carlo and quasi Monte Carlo methods, to be able to construct Monte Carlo methods with variance reduction using antithetic variates and control variates; write computer programs for that;
· Describe the maximum likelihood method for parameter estimation in continuous time models used for option pricing, perform asymptotic analysis of the estimates in case of high frequency data, as well as in case of a long time series of data;
· Construct the Monte Carlo methods for pricing in stochastic volatility models.

Assessment: Total Marks 100: End of Year Written Examination 80 marks; Continuous Assessment 20 marks (one in-class test: 10 marks; Homework assignments 10 marks (3 x 3.33 marks each)).

Compulsory Elements: End of Year Written Examination; Continuous Assessment.

Penalties (for late submission of Course/Project Work etc.): Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

Pass Standard and any Special Requirements for Passing Module: 40%.

End of Year Written Examination Profile: 1 x 1½ hr(s) paper(s).

Requirements for Supplemental Examination: 1 x 1½ hr(s) paper(s) to be taken in Autumn. The mark for Continuous Assessment is carried forward.

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MF4053 Project

Credit Weighting: 5

Teaching Period(s): Teaching Periods 1 and 2.

No. of Students: Min 1, Max 30.

Pre-requisite(s): Pass in Year 3 of Financial Mathematics and Actuarial Science, or its equivalent

Co-requisite(s): None

Teaching Methods: Directed Study (Directed Reading; Individual Research; Computer Analysis; Presentation of Findings).

Module Co-ordinator: Professor Bernard Hanzon, School of Mathematical Sciences (Head of Department of Mathematics).

Lecturer(s): Dr Grigory Temnov, School of Mathematical Sciences.

Module Objective: To develop skills in independent study and research in the Mathematical Sciences relevant to Financial Mathematics and Actuarial Science.

Module Content: Participants will conduct an investigative project in Financial Mathematics or Actuarial Science, or in a relevant area of Mathematical Sciences.

Learning Outcomes: On successful completion of this module, students should be able to:
· Write a logical and coherent account on a mathematical or statistical topic that is related to the field of financial mathematics and actuarial science;
· Present and analyze results from the literature;
· Carry out independent literature research;
· Construct proofs of mathematical results and construct algorithms for the calculation of relevant quantities;

Assessment: Total Marks 100: Continuous Assessment 100 marks (Project Report to be submitted by the end of lectures in Period 2).

Compulsory Elements: Continuous Assessment.

Penalties (for late submission of Course/Project Work etc.): Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

Pass Standard and any Special Requirements for Passing Module: 40%.

End of Year Written Examination Profile: No End of Year Written Examination.

Requirements for Supplemental Examination: Marks in passed element(s) of Continuous Assessment are carried forward, Failed element(s) of Continuous Assessment must be repeated (Resubmission of revised project, as prescribed by the School).

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MF4054 Stochastic Modelling II

Credit Weighting: 5

Teaching Period(s): Teaching Period 2.

No. of Students: Min 5, Max 30.

Pre-requisite(s): ST3053, MA3051, AM3063, MA4058

Co-requisite(s): None

Teaching Methods: 24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

Module Co-ordinator: Professor Bernard Hanzon, School of Mathematical Sciences.

Lecturer(s): Dr Claus Michael Koestler, School of Mathematical Sciences.

Module Objective: To teach the theory and applications of continuous time parameter stochastic processes.

Module Content: Continuous time stochastic processes: Gaussian random vectors, Brownian motion, continuous time martingales, stochastic diffusions. Stochastic calculus: introduction to Ito calculus, introduction to stochastic differential equations, Ito calculus and PDEs.

Learning Outcomes: On successful completion of this module, students should be able to:
· Demonstrate an understanding of the basic concepts and theorems of Gaussian random vectors and their use in the construction of Brownian motion;
· Discuss Brownian motion with drift and Geometric Brownian motion and mean-reverting processes;
· Synthesise techniques that have been developed in the course to solve particular problems;
· Explain the basic concepts and the construction of the Ito integral and Ito processes;
· Solve problems involving stochastic integration and elementary stochastic analysis;
· Explain and apply elementary Ito calculus;
· Evaluate basic Ito integrals.
· Feyman-Kac theorem; Mean reverting processes; Ornstein-Uhlenbeck models.

Assessment: Total Marks 100: End of Year Written Examination 80 marks; Continuous Assessment 20 marks (one in-class test: 10 marks; Homework assignments 10 marks (2 x 5 marks each)).

Compulsory Elements: End of Year Written Examination; Continuous Assessment.

Penalties (for late submission of Course/Project Work etc.): None.

Pass Standard and any Special Requirements for Passing Module: 40%.

End of Year Written Examination Profile: 1 x 1½ hr(s) paper(s).

Requirements for Supplemental Examination: 1 x 1½ hr(s) paper(s) to be taken in Autumn.

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MF4055 Statistical Methods in Insurance

Credit Weighting: 5

Teaching Period(s): Teaching Period 2.

No. of Students: Min 5, Max 100.

Pre-requisite(s): ST2054, ST3051

Co-requisite(s): None

Teaching Methods: 24 x 1hr(s) Lectures; 10 x 1hr(s) Tutorials.

Module Co-ordinator: Prof Finbarr O'Sullivan, Department of Statistics.

Lecturer(s): Mr Damian Conway, Department of Statistics.

Module Objective: To provide instruction in the elements of Statistical methods relevant to actuarial work, particularly in non-life insurance.

Module Content: Loss distributions; Reinsurance; Compound distributions; Risk models; Ruin theory; Experience rating; Estimating the ultimate cost of claims: run-off triangles; Credibility Theory.

Learning Outcomes: On successful completion of this module, students should be able to:
· Describe the operation of simple forms of reinsurance contracts.
· Describe the properties of the main statistical (loss) distributions used to model individual and aggregate insurance claims and be able to derive moments of such loss distributions used in the modelling of insurance claims data.
· Estimate the parameters of a loss distribution when the data is complete or when it is incomplete, using maximum likelihood and the method of moments.
· Describe the main characteristics of the more common forms of general insurance products.
· Describe the basic methodology an insurer would use in rating various classes of insurance business.
· Explain the concepts of Ruin Theory and how these can be applied to Risk Models to analyse the solvency of an insurance company.
· Define a compound Poisson / Binomial Process and illustrate how such processes can be used to construct a model of aggregate insurance claims.
· Explain the empirical Bayes approach to credibility theory and calculate credibility premium estimates for such models.
· Describe the concepts of reserving in relation to insurance business and be able to perform a number of techniques for the calculation of reserves to be held under short tailed insurance contacts.

Assessment: Total Marks 100: End of Year Written Examination 80 marks; Continuous Assessment 20 marks (1 x in-class test 12 marks, 1 x homework 8 marks).

Compulsory Elements: End of Year Written Examination; Continuous Assessment.

Penalties (for late submission of Course/Project Work etc.): Work which is submitted late shall be assigned a mark of zero (or a Fail Judgement in the case of Pass/Fail modules).

Pass Standard and any Special Requirements for Passing Module: 40%.

End of Year Written Examination Profile: 1 x 1½ hr(s) paper(s).

Requirements for Supplemental Examination: 1 x 1½ hr(s) paper(s) to be taken in Autumn. The mark for Continuous Assessment is carried forward.

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